Conventional knowledge suggests that pickleball paddles behave like tennis racquets, which have strings that are supported by a rigid outer “frame”. When striking the ball, the strings stretch and deform like a trampoline, thereby storing the kinetic energy of the ball in the form of potential energy in the strings. At maximum deformation, the potential energy in the strings is released and transferred back to the ball, which increases its velocity on rebound. This transformation of energy is known as the “trampoline effect”.
In the sport of baseball, it is well known that traditional solid wooden baseball bats do not exhibit a trampoline effect. That is, the strain energy stored in the wooden bat is not effectively returned to the ball. This is not the case for modern hollow metal and composite bats, where energy is stored in the so-called hoop modes, which can efficiently return energy to the ball. This can result in “hotter” bats with significantly more power than traditional solid wood bats. Consequently, hollow metal and composite bats are not allowed in professional baseball.
Because of the greater potential for injury by being hit by batted balls and the over-abundance of homeruns at the high school and collegiate levels, governing bodies (such as the NFHS and NCAA) regulate bats to have a Coefficient of Restitution (or CoR) of no greater than 0.50 as measured by the Baseball / Bat Coefficient of Restitution test (BBCoR). Interestingly, in the sport of tennis, governing bodies (such as the International Tennis Federation, or ITF) do not limit the CoR of tennis racquets but instead focus on rules related to the racquet’s physical dimensions and the hitting surface. In the sport of pickleball, the USAP has adopted a form of the BBCoR test that they call the Paddle / Ball Coefficient of Restitution test (or PBCoR) to limit the power or reactivity of pickleball paddles. Is this really necessary? Do pickleball paddles behave more like baseball bats or tennis racquets? To answer these questions, let’s look at the dynamic interactions between the pickleball and the paddle.
Dynamic Interactions
In a previous article, we simulated the impact of a ball at the center of a paddle using an impulse hammer and a weighted accelerometer. For the Ronbus Ripple V1 paddle, the so-called driving point transfer function, which is the dynamic response of the paddle at the ball strike location, is shown in Figure 1. This indicates that when the ball strikes the paddle, it will excite two vibration modes: the diving board mode at about 394 Hz and the trampoline mode at about 492 Hz.
Figure 2a shows that the duration of the impulse is about 3.3 msec in the shape of a half-sine wave. Although the peak force level was about 30 lbs, we normalized the impulse and response data to a unit force. The response (Figure 2b) shows that during contact the paddle trampoline mode at around 500 Hz is being excited. It is interesting to note that even during the contact duration of 3.3 msec the paddle vibrates. After contact, we see residual “ringing” of the paddle at around 400 Hz, corresponding to the diving board mode.
Paddle Equivalent Mass & Stiffness
Assuming that the paddles behave like tennis racquets, we can use the dynamic data to develop an equivalent spring/mass system. Knowing the paddle trampoline vibration frequency, it is now possible to estimate the dynamic mass (m) and stiffness (k) of the paddle by using the well-known natural frequency equation:
In our impulse test, we assumed that the ball acts like a rigid mass, which was simulated by using an accelerometer that was weighted to 1 oz. Assuming that the effective mass of the paddle is about one-quarter of the total paddle mass (8 oz), the total effective mass would be 3 oz (0.085 kg). For the trampoline mode at around 500 Hz, we calculate that the effective stiffness of this vibration mode must be 839 kN/m (4,790 lb/in).
In a previous article (see “Paddle Power vs Control Comparison”), we created a three-point bending test apparatus which essentially constrains paddles along their lateral edges and throat. The three-point bending test was designed to calculate the paddle face stiffness, assuming that it behaved like a tennis racquet, with a rigid edge “frame” and a compliant center. According to the test data, the Ronbus Ripple v1 has a face stiffness of 609 lb/in. This is almost one-eighth the stiffness calculated by the impulse dynamic test (4,790 lb/in)! What gives? Are the tests and calculations wrong?
Further interpretation suggests that we are looking at the paddle static and dynamic deformations incorrectly. Force does not appear to be transferred from the handle to the edge to the paddle face in the same manner as a tennis racquet. Instead, the forward momentum of the paddle center of mass causes the paddle edges to deform around the ball at the instant of contact. The differences between these loading conditions can be illustrated in an examination of boundary conditions.
Boundary Conditions
Assuming that we can consider the paddle to behave like a beam and the ball to behave like a concentrated mass, Figure 3 shows how a tennis racquet would be modeled, with the frame constraining the face at the edges and the mass of the ball concentrated at the center of the beam. Here, the paddle face provides an effective bending stiffness (k) against which the effective mass (m) oscillates at the trampoline mode frequency. The effective mass includes the mass of the ball plus a portion of the mass of the paddle face. In this case, dynamic loads are reacted through the pin-pin edge constraints.
Figure 3. Pin-Pin Boundary Conditions
Figure 4 shows how the pickleball paddle would behave under free-free boundary conditions. Although the shape and frequency of the trampoline modes are identical for the pin-pin and free-free boundary conditions, there is a subtle difference in how the dynamic loads are reacted. For the pin-pin boundary conditions shown in Figure 3, the dynamic loads are reacted through the supports (or edge of the paddle). For the free-free boundary conditions shown in Figure 4, the dynamic loads are reacted through the paddle/ball center of mass. Essentially, the center of mass behaves like an instantaneous center or nodal point of vibration, where the edges of the paddle are just “flapping” about the center of mass in response to the impulse load from the ball.
Figure 4. Free-Free Boundary Conditions
Conclusions
This analysis suggests that although a pickleball paddle exhibits trampoline-like behavior, the edge of the paddle might not constrain the paddle face in the same manner that a tennis racquet frame constrains the strings. Therefore, the conventional knowledge that the pickleball paddle behaves like a tennis racquet, where the trampoline effect of the paddle face helps launch the ball on rebound might not be correct.
These results are important because several static, dynamic, and impact tests have been used to certify that the trampoline effects of certain paddles do not provide players using them with an unfair advantage. If the trampoline effect is not an appropriate indicator of paddle power, there is a possibility that paddles that would otherwise be “legal” are disqualified, and vice versa.
In our next article, we will examine whether the dynamic behavior of a pickleball paddle is like that of a baseball bat and determine how the paddle dynamics may contribute to or detract from the performance of the paddle.
